WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sel(0(),cons(X,Z)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Signature: {activate/1,first/2,from/1,s/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from,s,sel} and constructors {0,cons ,n__first,n__from,n__s,nil} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) All above mentioned rules can be savely removed. * Step 2: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sel(0(),cons(X,Z)) -> X - Signature: {activate/1,first/2,from/1,s/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from,s,sel} and constructors {0,cons ,n__first,n__from,n__s,nil} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(first) = {1,2}, uargs(from) = {1}, uargs(s) = {1} Following symbols are considered usable: {activate,first,from,s,sel} TcT has computed the following interpretation: p(0) = [1] p(activate) = [1] x1 + [0] p(cons) = [1] x1 + [3] p(first) = [1] x1 + [1] x2 + [6] p(from) = [1] x1 + [4] p(n__first) = [1] x1 + [1] x2 + [6] p(n__from) = [1] x1 + [4] p(n__s) = [1] x1 + [8] p(nil) = [7] p(s) = [1] x1 + [8] p(sel) = [1] x1 + [4] x2 + [1] Following rules are strictly oriented: from(X) = [1] X + [4] > [1] X + [3] = cons(X,n__from(n__s(X))) sel(0(),cons(X,Z)) = [4] X + [14] > [1] X + [0] = X Following rules are (at-least) weakly oriented: activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__first(X1,X2)) = [1] X1 + [1] X2 + [6] >= [1] X1 + [1] X2 + [6] = first(activate(X1),activate(X2)) activate(n__from(X)) = [1] X + [4] >= [1] X + [4] = from(activate(X)) activate(n__s(X)) = [1] X + [8] >= [1] X + [8] = s(activate(X)) first(X1,X2) = [1] X1 + [1] X2 + [6] >= [1] X1 + [1] X2 + [6] = n__first(X1,X2) first(0(),Z) = [1] Z + [7] >= [7] = nil() from(X) = [1] X + [4] >= [1] X + [4] = n__from(X) s(X) = [1] X + [8] >= [1] X + [8] = n__s(X) * Step 3: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() from(X) -> n__from(X) s(X) -> n__s(X) - Weak TRS: from(X) -> cons(X,n__from(n__s(X))) sel(0(),cons(X,Z)) -> X - Signature: {activate/1,first/2,from/1,s/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from,s,sel} and constructors {0,cons ,n__first,n__from,n__s,nil} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(first) = {1,2}, uargs(from) = {1}, uargs(s) = {1} Following symbols are considered usable: {activate,first,from,s,sel} TcT has computed the following interpretation: p(0) = [0] p(activate) = [2] x1 + [0] p(cons) = [1] x1 + [0] p(first) = [1] x1 + [1] x2 + [3] p(from) = [1] x1 + [10] p(n__first) = [1] x1 + [1] x2 + [3] p(n__from) = [1] x1 + [6] p(n__s) = [1] x1 + [3] p(nil) = [0] p(s) = [1] x1 + [6] p(sel) = [1] x1 + [8] x2 + [5] Following rules are strictly oriented: activate(n__first(X1,X2)) = [2] X1 + [2] X2 + [6] > [2] X1 + [2] X2 + [3] = first(activate(X1),activate(X2)) activate(n__from(X)) = [2] X + [12] > [2] X + [10] = from(activate(X)) first(0(),Z) = [1] Z + [3] > [0] = nil() from(X) = [1] X + [10] > [1] X + [6] = n__from(X) s(X) = [1] X + [6] > [1] X + [3] = n__s(X) Following rules are (at-least) weakly oriented: activate(X) = [2] X + [0] >= [1] X + [0] = X activate(n__s(X)) = [2] X + [6] >= [2] X + [6] = s(activate(X)) first(X1,X2) = [1] X1 + [1] X2 + [3] >= [1] X1 + [1] X2 + [3] = n__first(X1,X2) from(X) = [1] X + [10] >= [1] X + [0] = cons(X,n__from(n__s(X))) sel(0(),cons(X,Z)) = [8] X + [5] >= [1] X + [0] = X * Step 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__s(X)) -> s(activate(X)) first(X1,X2) -> n__first(X1,X2) - Weak TRS: activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) first(0(),Z) -> nil() from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sel(0(),cons(X,Z)) -> X - Signature: {activate/1,first/2,from/1,s/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from,s,sel} and constructors {0,cons ,n__first,n__from,n__s,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(first) = {1,2}, uargs(from) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [3] x1 + [0] p(cons) = [1] x1 + [0] p(first) = [1] x1 + [1] x2 + [0] p(from) = [1] x1 + [0] p(n__first) = [1] x1 + [1] x2 + [0] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [1] p(nil) = [0] p(s) = [1] x1 + [2] p(sel) = [1] x2 + [0] Following rules are strictly oriented: activate(n__s(X)) = [3] X + [3] > [3] X + [2] = s(activate(X)) Following rules are (at-least) weakly oriented: activate(X) = [3] X + [0] >= [1] X + [0] = X activate(n__first(X1,X2)) = [3] X1 + [3] X2 + [0] >= [3] X1 + [3] X2 + [0] = first(activate(X1),activate(X2)) activate(n__from(X)) = [3] X + [0] >= [3] X + [0] = from(activate(X)) first(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__first(X1,X2) first(0(),Z) = [1] Z + [0] >= [0] = nil() from(X) = [1] X + [0] >= [1] X + [0] = cons(X,n__from(n__s(X))) from(X) = [1] X + [0] >= [1] X + [0] = n__from(X) s(X) = [1] X + [2] >= [1] X + [1] = n__s(X) sel(0(),cons(X,Z)) = [1] X + [0] >= [1] X + [0] = X Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X first(X1,X2) -> n__first(X1,X2) - Weak TRS: activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) first(0(),Z) -> nil() from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sel(0(),cons(X,Z)) -> X - Signature: {activate/1,first/2,from/1,s/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from,s,sel} and constructors {0,cons ,n__first,n__from,n__s,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(first) = {1,2}, uargs(from) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(activate) = [2] x1 + [0] p(cons) = [1] x1 + [0] p(first) = [1] x1 + [1] x2 + [8] p(from) = [1] x1 + [0] p(n__first) = [1] x1 + [1] x2 + [4] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [0] p(nil) = [9] p(s) = [1] x1 + [0] p(sel) = [1] x2 + [0] Following rules are strictly oriented: first(X1,X2) = [1] X1 + [1] X2 + [8] > [1] X1 + [1] X2 + [4] = n__first(X1,X2) Following rules are (at-least) weakly oriented: activate(X) = [2] X + [0] >= [1] X + [0] = X activate(n__first(X1,X2)) = [2] X1 + [2] X2 + [8] >= [2] X1 + [2] X2 + [8] = first(activate(X1),activate(X2)) activate(n__from(X)) = [2] X + [0] >= [2] X + [0] = from(activate(X)) activate(n__s(X)) = [2] X + [0] >= [2] X + [0] = s(activate(X)) first(0(),Z) = [1] Z + [9] >= [9] = nil() from(X) = [1] X + [0] >= [1] X + [0] = cons(X,n__from(n__s(X))) from(X) = [1] X + [0] >= [1] X + [0] = n__from(X) s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) sel(0(),cons(X,Z)) = [1] X + [0] >= [1] X + [0] = X Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X - Weak TRS: activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sel(0(),cons(X,Z)) -> X - Signature: {activate/1,first/2,from/1,s/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from,s,sel} and constructors {0,cons ,n__first,n__from,n__s,nil} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(first) = {1,2}, uargs(from) = {1}, uargs(s) = {1} Following symbols are considered usable: {activate,first,from,s,sel} TcT has computed the following interpretation: p(0) = [0] p(activate) = [4] x1 + [8] p(cons) = [1] x1 + [0] p(first) = [1] x1 + [1] x2 + [8] p(from) = [1] x1 + [1] p(n__first) = [1] x1 + [1] x2 + [4] p(n__from) = [1] x1 + [1] p(n__s) = [1] x1 + [4] p(nil) = [8] p(s) = [1] x1 + [5] p(sel) = [4] x1 + [1] x2 + [2] Following rules are strictly oriented: activate(X) = [4] X + [8] > [1] X + [0] = X Following rules are (at-least) weakly oriented: activate(n__first(X1,X2)) = [4] X1 + [4] X2 + [24] >= [4] X1 + [4] X2 + [24] = first(activate(X1),activate(X2)) activate(n__from(X)) = [4] X + [12] >= [4] X + [9] = from(activate(X)) activate(n__s(X)) = [4] X + [24] >= [4] X + [13] = s(activate(X)) first(X1,X2) = [1] X1 + [1] X2 + [8] >= [1] X1 + [1] X2 + [4] = n__first(X1,X2) first(0(),Z) = [1] Z + [8] >= [8] = nil() from(X) = [1] X + [1] >= [1] X + [0] = cons(X,n__from(n__s(X))) from(X) = [1] X + [1] >= [1] X + [1] = n__from(X) s(X) = [1] X + [5] >= [1] X + [4] = n__s(X) sel(0(),cons(X,Z)) = [1] X + [2] >= [1] X + [0] = X * Step 7: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) first(X1,X2) -> n__first(X1,X2) first(0(),Z) -> nil() from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) sel(0(),cons(X,Z)) -> X - Signature: {activate/1,first/2,from/1,s/1,sel/2} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from,s,sel} and constructors {0,cons ,n__first,n__from,n__s,nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))